The arithmetic mean of two positive numbers is 10 and their geometric mean is 8. The numbers are:

The conditions give $a+b=20$ and $ab=64$, so a and $b$ are the roots of $t^2-20t+64=0$. Since √$(400-256)=12$, the roots are (20 ± 12)/2 = 16 and 4. Check: their mean is 10 and √$(4\times 16)=8$. (8 and 12 have the right arithmetic mean but not the right geometric mean.)